Mathematics and Its History pp 415-437 | Cite as

# Hypercomplex Numbers

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This chapter is the story of a generalization with an unexpected outcome. In trying to generalize the concept of real number to *n* dimensions, we find only four dimensions where the idea works: *n* = 1, 2, 4, 8. “Numberlike” behavior in ℝ*n*, far from being common, is a rare and interesting exception. Our idea of “numberlike” behavior is motivated by the cases *n* = 1, 2 that we already know: the real numbers ℝ and the complex numbers ℂ. The number systems ℝ and ℂ have both algebraic and geometric properties in common. The common algebraic property is that of being a *field*, and it is captured by nine laws governing addition and multiplication, such as *ab* = *ba* and *a*(*bc*) = (*ab*)*c* (commutative and associative laws for multiplication). The common geometric property is the existence of an *absolute value*, |*u*|, which measures the distance of *u* from *O* and is *multiplicative*: |*uv*| = |*u*||*v*|. In the 1830s and 1840s, Hamilton and Graves searched long and hard for “numberlike” behavior in ℝ_{ n }, but they came up short. Beyond ℝ and ℂ, only two *hypercomplex number* systems even come close: for *n* = 4 the *quaternion* algebra ℍ, which has all the required properties except commutative multiplication, and for *n* = 8 the *octonion* algebra \(\mathbb{O}\), which has all the required properties except commutative and associative multiplication. Despite lacking some of the field properties, ℍ and \(\mathbb{O}\) can serve as coordinates for projective planes. In this setting, the missing field properties have a remarkable geometric meaning. Failure of the commutative law corresponds to failure of the Pappus theorem, and failure of the associative law corresponds to failure of the Desargues theorem.

## Keywords

Projective Plane Multiplicative Property Commutative Multiplication Octonion Algebra Hypercomplex Number## Preview

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